HOW TO DO THIS WHOLE QUESTION. HELP! A manufacturer produces x% of defective lig

Question

HOW TO DO THIS WHOLE QUESTION. HELP!

A manufacturer produces x% of defective light bulbs on the average. Each item is inspected before being put on the market. Items that are classified as good are put on the market and those classified as defective are scrapped. In the past, the inspectors misclassify an item 5% of the time. After inspecting all the light bulbs, it is found that 77% of items are classified as good and put on the market. (a) Find the percentage of light bulbs that are actually defective. (b) A bulb is selected at random from those that are put on the market. Find the probability that the item is actually good

Explanation / Answer

Let

G = bulb is classified as good
D = bulb is actually defective

a)

Note that

P(G) = P(D) P(G|D) + P(D') P(G|D')

0.77 = P(D) (0.05) + (1-P(D))(0.95)

0.77 = -0.90P(D) + 0.95

P(D) = 0.20 [ANSWER]

***********************

b)

Here,

P(D'|G) = P(D') P(G|D')/P(G)

= (1-0.20)*(1-0.05)/0.77 = 0.987012987 [ANSWER]

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